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Properties

Label 1-23e2-529.225-r0-0-0
Degree $1$
Conductor $529$
Sign $0.669 + 0.742i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.926 + 0.375i)2-s + (0.890 + 0.454i)3-s + (0.717 − 0.696i)4-s + (0.912 + 0.409i)5-s + (−0.996 − 0.0868i)6-s + (−0.492 − 0.870i)7-s + (−0.404 + 0.914i)8-s + (0.586 + 0.809i)9-s + (−0.999 − 0.0372i)10-s + (0.481 + 0.876i)11-s + (0.955 − 0.293i)12-s + (0.105 − 0.994i)13-s + (0.783 + 0.621i)14-s + (0.626 + 0.779i)15-s + (0.0310 − 0.999i)16-s + (0.948 − 0.317i)17-s + ⋯
L(s)  = 1  + (−0.926 + 0.375i)2-s + (0.890 + 0.454i)3-s + (0.717 − 0.696i)4-s + (0.912 + 0.409i)5-s + (−0.996 − 0.0868i)6-s + (−0.492 − 0.870i)7-s + (−0.404 + 0.914i)8-s + (0.586 + 0.809i)9-s + (−0.999 − 0.0372i)10-s + (0.481 + 0.876i)11-s + (0.955 − 0.293i)12-s + (0.105 − 0.994i)13-s + (0.783 + 0.621i)14-s + (0.626 + 0.779i)15-s + (0.0310 − 0.999i)16-s + (0.948 − 0.317i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.669 + 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346896082 + 0.5989217896i\)
\(L(\frac12)\) \(\approx\) \(1.346896082 + 0.5989217896i\)
\(L(1)\) \(\approx\) \(1.079602167 + 0.3230930660i\)
\(L(1)\) \(\approx\) \(1.079602167 + 0.3230930660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.926 + 0.375i)T \)
3 \( 1 + (0.890 + 0.454i)T \)
5 \( 1 + (0.912 + 0.409i)T \)
7 \( 1 + (-0.492 - 0.870i)T \)
11 \( 1 + (0.481 + 0.876i)T \)
13 \( 1 + (0.105 - 0.994i)T \)
17 \( 1 + (0.948 - 0.317i)T \)
19 \( 1 + (-0.982 - 0.185i)T \)
29 \( 1 + (0.879 + 0.476i)T \)
31 \( 1 + (-0.998 + 0.0620i)T \)
37 \( 1 + (0.922 + 0.386i)T \)
41 \( 1 + (0.767 + 0.640i)T \)
43 \( 1 + (0.392 - 0.919i)T \)
47 \( 1 + (0.203 - 0.979i)T \)
53 \( 1 + (-0.999 + 0.0372i)T \)
59 \( 1 + (-0.239 + 0.970i)T \)
61 \( 1 + (0.275 - 0.961i)T \)
67 \( 1 + (-0.596 + 0.802i)T \)
71 \( 1 + (-0.986 - 0.160i)T \)
73 \( 1 + (0.879 - 0.476i)T \)
79 \( 1 + (-0.166 - 0.985i)T \)
83 \( 1 + (0.988 - 0.148i)T \)
89 \( 1 + (-0.944 - 0.329i)T \)
97 \( 1 + (0.992 - 0.123i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.76559758040420553177857184929, −22.05485256842723209665144685113, −21.25127800122407489222382306265, −21.03719320576329148027008370544, −19.7012620733089225144144911921, −19.12800489551089567800836185533, −18.57808604949385244354323070676, −17.64776308397263333236610666450, −16.65777508576634659245980422451, −16.02058450783569119428270433108, −14.73708206117605407039741350000, −13.93926059945312106839351213675, −12.78610800487939595710917296324, −12.376434217727159706524417997500, −11.19562106511783807346699644104, −9.92128812142498298491292675355, −9.21485350668141551256957079245, −8.75196465732412256991454792054, −7.84503386287156941096580871973, −6.47137122234390298738957724486, −6.02020149369022739842221034652, −4.03781585949421592718948742172, −2.90342733076556705456832192511, −2.08695628526770151964714737624, −1.162163476534824464900902923433, 1.275152491719786902852750477155, 2.41005208532202393125325471821, 3.37163171080266876036542367977, 4.79539170108287993180557247742, 6.03862422660509168330524542159, 7.04748247512955616515800312794, 7.7148328592180111310577137949, 8.877151403396331555494490589782, 9.69959952494540521002042246656, 10.244119717815643981908743653678, 10.84694883270559114634801673041, 12.57654143242378754923750206231, 13.56108918629070472193340755096, 14.529830850856473400254472506160, 14.94602068753307714541716913969, 16.075097953694204628621199417338, 16.87654573271278987568348739031, 17.62775200014308464919992953610, 18.52031069740476978646751533918, 19.45670004802567390018466433566, 20.17254461133458382424031699511, 20.73451414973829971239814313781, 21.768567418850001217436688260983, 22.8176203938966422524713747035, 23.682253523438149834284607211271

Graph of the $Z$-function along the critical line